In that way we have generalized the expression for the quantum hydrodynamic propagator given in the de Broglie-Bohm theory [1]}. Notice two alternative gauge-independent expressions are given. These are formulas relating the wavefunction \(\Phi \)  at two different times and spatial locations, through an exponential factor that involves integration over past times. It would be interesting to find a relation between \(\Phi \)  and \(\Psi \)  at the same time and location. In order to do this we need to make an assumption about the phase. We assume that the phase of the unperturbed system does not change significantly when we include the non-linear term. It is reasonable to assume that the imaginary term will more appreciably affect the evolution equation of the probability density and to a lesser extent the time development of the phase since the Hamilton-Jacobi that governs the evolution of the phase equation will remain unaltered. Certainly, we can add correction terms to this initial phase and use (5) to obtain iteratively better approximate expressions. Using (8) and setting \(\varepsilon =0\)  we find
\(A(\mathbf {r},t)=e^{-\frac{1}{2m}\int _{t_{0}}^{t}d\tau \nabla ^{2}S}A(\mathbf {r},t_{0}),\) 
